Import Question JSON

Current Question (ID: 9810)

Question:

$\text{A point mass } m \text{ is moved in a vertical circle of radius } r \text{ with the help of a string. The velocity of the mass is } \sqrt{7gr} \text{ at the lowest point. The tension in the string at the lowest point will be:}$

Options:

1. $6mg$
2. $7mg$
3. $8mg$
4. $mg$

Solution:

\text{Hint: Balance all the forces in the vertical direction.} \text{Step 1: Draw the FBD of the given mass.} \text{At the lowest point of the vertical circle, the forces acting on the mass are:} \text{- Tension } T \text{ acting upward along the string} \text{- Weight } mg \text{ acting downward} \text{- The mass has velocity } v = \sqrt{7gr} \text{ horizontally} \text{Step 2: Find the tension in the string} \text{Applying Newton's second law in the radial direction (upward positive):} T - mg = \frac{mv^2}{r} \text{Since } v = \sqrt{7gr}\text{, we have } v^2 = 7gr \text{Therefore: } \frac{mv^2}{r} = \frac{m \cdot 7gr}{r} = 7mg T - mg = 7mg T = 8mg \text{Hence, option (3) is the correct answer.}

Import JSON File

Upload a JSON file containing LaTeX/MathJax formatted question, options, and solution.

Expected JSON Format:

{
  "question": "The mass of carbon present in 0.5 mole of $\\mathrm{K}_4[\\mathrm{Fe(CN)}_6]$ is:",
  "options": [
    {
      "id": 1,
      "text": "1.8 g"
    },
    {
      "id": 2,
      "text": "18 g"
    },
    {
      "id": 3,
      "text": "3.6 g"
    },
    {
      "id": 4,
      "text": "36 g"
    }
  ],
  "solution": "\\begin{align}\n&\\text{Hint: Mole concept}\\\\\n&1 \\text{ mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6] = 6 \\text{ moles of carbon atom}\\\\\n&0.5 \\text{ mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6] = 6 \\times 0.5 \\text{ mol} = 3 \\text{ mol}\\\\\n&1 \\text{ mol of carbon} = 12 \\text{ g}\\\\\n&3 \\text{ mol carbon} = 12 \\times 3 = 36 \\text{ g}\\\\\n&\\text{Hence, 36 g mass of carbon present in 0.5 mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6].\n\\end{align}",
  "correct_answer": 4
}